First, I know I have to make an odd or even extension to the whole line. But for both even and odd extensions, the formulae are:
u(x,t)=1/(√4∏kt)(∫(e-(x-y)2/4kt±e-(x+y)2/4kt)ψ(y)dy)
this means for u(x,0)=0, everything is multiplied by 0, which clearly isn't right.

Second, I don't know which extension to apply, it seems like both should work: I can either make an odd extension about the line x=1 or an even extension about the t axis, both will intercept the t axis at 1, hence satisfying the BC.